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Entailment with Conditional Equality Constraints

Entailment with Conditional Equality Constraints. Zhendong Su Alex Aiken University of California, Berkeley. Constraint Based Analysis. Basic Idea:. program source. constraint generation. constraints. constraint resolution. analysis results. Constraint Simplification.

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Entailment with Conditional Equality Constraints

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  1. Entailment with Conditional Equality Constraints Zhendong Su Alex Aiken University of California, Berkeley

  2. Constraint Based Analysis • Basic Idea: program source constraint generation constraints constraint resolution analysis results ESOP'01

  3. Constraint Simplification • Removing redundant constraints from a system. • Example (using unification constraints) • C = {x=y, y=z, x=z} • x=y and y=z implies x=z, get C’ = {x=y, y=z} • How about if we are only interested in the variables x and y? ESOP'01

  4. Constraint Entailment • Entailment is a decision problem • justify a potential candidate for simplification • shed light on the hardness of simplification • Simple Entailment C1  C2 • holds if every solution of C1 is a solution of C2. • Example: {x=y,y=z}  {x=z} • Restricted Entailment C1 V C2 • holds if for every solution of C1 there exists a solution of C2 s.t. they agree on V. • Example: {x=y} {x,y} {x=y,y=z,x=z} ESOP'01

  5. How to Use Entailment? • Simplify C • Simplify C with respect to V C’ := C for each ci  C if(C’\{ci}  ci) C’ := C’\{ci} C’ := C for each ci  C if(C’\{ci} V C’) C’ := C’\{ci} ESOP'01

  6. Related Work • Theoretical issues • subtyping entailment • (C1    ) Henglein and Rehof [LICS’97, ICALP’98] • set constraint entailment • (C1 V C2) Flanagan and Felleisen [PLDI’97] • atomic set constraint entailment • (C1    ) Niehren, Mueller, and Talbot [LICS’99] • polymorphic type simplification • (.\C) Aiken, Wimmers, and Palsberg [TACS’97] ESOP'01

  7. Related Work (cont.) • Practical simplifications • polymorphic type simplification • Fahndrich and Aiken [SC’96] • Pottier [ICFP’96] • Marlow and Wadler (Erlang) [ICFP’97] • set constraint simplification (C1 V C2) • Flanagan and Felleisen [PLDI’97] • constraint solving • Fahndrich, Foster, Su, and Aiken [PLDI’98] • Su, Fahndrich, and Aiken [POPL’00] ESOP'01

  8. Contributions • Give the first interesting class with efficiently decidable entailment problem. • Use novel techniques in constructing the algorithm. • Provide a natural boundary between tractable and intractable constraint theories. ESOP'01

  9. Conditional Unification • simple types:  ::=  |  |  | 12 • ground types: simple types without variables • constraints: 1 = 2   • valuation:  :    •  : variables  : ground types • satisfaction: •  1 = 2  (1)= (2) •      () =  or () = () ESOP'01

  10. Simple Entailment • Theorem 1. C1  C2 is decidable in polynomial time. • Basic idea to decide C   =  •      implies    •    and    implies  =  •     and   implies  =  • apply unification and congruence closure to check if  =  ESOP'01

  11. Restricted Entailment • Theorem 2. C1 V C2 is decidable in polynomial time. • The most interesting and difficult result of the paper. • The key idea: transform the constraints so that we only need to consider at most two conditional constraints at a time. ESOP'01

  12. An Example       t1 t2 t3 t C2 C1 Does C1 C2 ? NO Does C1 {, ,} C2 ? YES ESOP'01

  13. Outline of the Algorithm • Introduce closed systems, for which it is sufficient to consider only pairs of conditional constraints • Entailment with pair constraints can be decided in polynomial time • Reduce entailment to entailment in terms of closed systems to get quadratic # of entailments of pair constraints ESOP'01

  14. Closed Systems           t1  t2  t3 t4 1 t1  t2  1 t t Example Closed system for example ESOP'01

  15. Closed Systems (cont.) • A property: it suffices to consider pairs of conditional constraints for the solutions of a closed system w.r.t. to a set of variables V.       t3 t4 1 t1  t2  1 t ESOP'01

  16. Pair Constraint Entailment • Lemma 1. C1 V C2 can be decidable in polynomial time if C2 consists only unification constraints . • Lemma 2. C1 V C2 can be decidable in polynomial time if C2 consists at most two conditional constraints . ESOP'01

  17. Completion • Lemma 3. Through a completion procedure, C1 V C2 can be reduced to C’1 V’ C’2, where C’2 is a closed system. • See paper for details. ESOP'01

  18. Putting Things Together • Main Theorem:C1 V C2 can be decide in polynomial time. • proof sketch: By Lemma 3, we reduce to entailment in terms of closed systems. We then consider quadratic # of entailment of pair constraints, where each one can be decided in polynomial time by Lemma 2. ESOP'01

  19. An Extension • A natural extension for comparison • add   (1 = 2) • intuitively means either  =  or 1 = 2 •       (= ) • Theorem 3. The entailment problem C1 V C2 is coNP-complete (reduction from 3SAT). ESOP'01

  20. The Reduction • 3SAT •  = C1 …  Cm • Ci = x  ¬y  z • associate x with x and ¬x • meaning x is true iff x and ¬x=    (1 = 2) Use for 2 1 ESOP'01

  21. The Reduction (cont.) • For each boolean variable x • For each clause Ci = x  ¬y  z x ¬x C1: part 1   t ¬ x y ¬z C1: part 2  s t  ESOP'01

  22. The Reduction (cont.) • To ensure at most one of x and¬x is •  is notsatisfiable iff C1VC2 (x, ¬x in V) x1 ¬x1 xn ¬xn ... xn t1 ¬x1 ... tn ¬xn x1 ...  s1 sn-1  ESOP'01

  23. Summary • Polynomial time algorithms for entailment • Algorithms can be used in practice to scale analyses based on conditional equality constraints • An extension to be used as a natural boundary between tractable and intractable constraint theories ESOP'01

  24. Open Problems • Many open problems in this area • Two long standing ones are: • Non-structural subtype entailment • Subtyping polymorphically constrained types • Hardness of simplification means good heuristics are very valuable ESOP'01

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